Frobenius theorem (real division algebras)

Proof

The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.

Introducing some notation

• Let D be the division algebra in question.
• Let n be the dimension of D.
• We identify the real multiples of 1 with R.
• When we write a ≤ 0 for an element a of D, we imply that a is contained in R.
• We can consider D as a finite-dimensional R-vector space. Any element d of D defines an endomorphism of D by left-multiplication, we identify d with that endomorphism. Therefore, we can speak about the trace of d, and its characteristic- and minimal polynomials.
• For any z in C define the following real quadratic polynomial:

:Note that if z ∈ C ∖ R then Q(z; x) is irreducible over R.

The claim

The key to the argument is the following

:Claim. The set V of all elements a of D such that a2 ≤ 0 is a vector subspace of D of dimension n − 1. Moreover as R-vector spaces, which implies that V generates D as an algebra.

Proof of Claim: Pick a in D with characteristic polynomial p(x). By the fundamental theorem of algebra, we can write

:p(x) = (x-t_1)\cdots(x-t_r) (x-z_1)(x - \overline{z_1}) \cdots (x-z_s)(x - \overline{z_s}), \qquad t_i \in \mathbf{R}, \quad z_j \in \mathbf{C} \setminus \mathbf{R}.

We can rewrite p(x) in terms of the polynomials Q(z; x):

:p(x) = (x-t_1)\cdots(x-t_r) Q(z_1; x) \cdots Q(z_s; x).

Since zj ∈ C ∖ R, the polynomials Q(zj; x) are all irreducible over R. By the Cayley–Hamilton theorem, and because D is a division algebra, it follows that either for some i or that for some j. The first case implies that a is real. In the second case, it follows that Q(zj; x) is the minimal polynomial of a. Because p(x) has the same complex roots as the minimal polynomial and because it is real it follows that

:p(x) = Q(z_j; x)^k = \left(x^2 - 2\operatorname{Re}(z_j) x + |z_j|^2 \right)^k

for some k. Since p(x) is the characteristic polynomial of a the coefficient of x 2k − 1 in p(x) is tr(a) up to a sign. Therefore, we read from the above equation we have: if and only if , in other words if and only if {{math|a2 −zj2

So V is the subset of all a with . In particular, it is a vector subspace. The rank–nullity theorem then implies that V has dimension n − 1 since it is the kernel of \operatorname{tr} : D \to \mathbf{R}. Since R and V are disjoint (i.e. they satisfy \mathbf R \cap V = {0}), and their dimensions sum to n, we have that .

The finish

For a, b in V define . Because of the identity , it follows that B(a, b) is real. Furthermore, since a2 ≤ 0, we have: B(a, a) 0 for a ≠ 0. Thus B is a positive-definite symmetric bilinear form, in other words, an inner product on V.

Let W be a subspace of V that generates D as an algebra and which is minimal with respect to this property. Let e1, ..., ek be an orthonormal basis of W with respect to B. Then orthonormality implies that:

:e_i^2 =-1, \quad e_i e_j = - e_j e_i.

The form of D then depends on k:

If , then D is isomorphic to R.

If , then D is generated by 1 and e1 subject to the relation . Hence it is isomorphic to C.

If , it has been shown above that D is generated by 1, e1, e2 subject to the relations :e_1^2 = e_2^2 =-1, \quad e_1 e_2 = - e_2 e_1, \quad (e_1 e_2)(e_1 e_2) =-1. These are precisely the relations for H.

If k 2, then D cannot be a division algebra. Assume that k 2. Define and consider . By rearranging the elements of this expression and applying the orthonormality relations among the basis elements we find that . If D were a division algebra, implies , which in turn means: and so e1, ..., e**k−1 generate D. This contradicts the minimality of W.

Remarks and related results

• The fact that D is generated by e1, ..., ek subject to the above relations means that D is the Clifford algebra of Rn. The last step shows that the only real Clifford algebras which are division algebras are Cℓ0, Cℓ1 and Cℓ2.
• As a consequence, the only commutative division algebras are R and C. Also note that H is not a C-algebra. If it were, then the center of H has to contain C, but the center of H is R.
• This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are R, C, H, and the (non-associative) algebra O.
• Pontryagin variant. If D is a connected, locally compact division ring, then , or H.

References

• Ray E. Artz (2009) Scalar Algebras and Quaternions, Theorem 7.1 "Frobenius Classification", page 26.
• Ferdinand Georg Frobenius (1878) "Über lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1–63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp. 343–405.
• Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp. 30–2 .
• Leonard Dickson (1914) Linear Algebras, Cambridge University Press. See §11 "Algebra of real quaternions; its unique place among algebras", pages 10 to 12.
• R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366–8.
• Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.